Personally I think that the restricted product description should be avoided. It is best to define $\widehat{\mathbb{Z}}$ to be the inverse limit of the system of all quotients $\mathbb{Z}/n$ (without gratuitously factoring $n$ as a product of primes) and then put $\mathbb{A}=(\mathbb{Q}\otimes\widehat{\mathbb{Z}})\times\mathbb{R}$. Now the adeles for any number field $K$ can be defined as $\mathbb{A}\otimes K$. The connection with primes/valuations for $K$ should be a theorem, not a definition.

Personally I think that the restricted product description should be avoided. It is best to define $\widehat{\mathbb{Z}}$ to be the inverse limit of the system of all quotients $\mathbb{Z}/n$ (without gratuitously factoring $n$ as a product of primes) and then put $\mathbb{A}=(\mathbb{Q}\otimes\widehat{\mathbb{Z}})\times\mathbb{R}$. We can topologise this by giving $\mathbb{R}$ the usual topology, and $\mathbb{Q}\otimes\widehat{\mathbb{Z}}$ the topology for which the sets $q\otimes\widehat{\mathbb{Z}}$ form a basis of neighbourhoods of zero. Now the adeles for any number field $K$ can be defined as $\mathbb{A}\otimes K$. Any $\mathbb{Q}$-basis for $K$ identifies $\mathbb{A}\otimes K$ with $\mathbb{A}^d$ and thus gives a topology on $\mathbb{A}\otimes K$, which is easily seen to be independent of the choice of basis. The connection with primes/valuations for $K$ should be a theorem, not a definition.